Explicit points on y2 + xy − tdy = x3 and related character sums

Let FqFq denote a finite field of characteristic p≥5p≥5 and let d=q+1d=q+1. Let EdEd denote the elliptic curve over the function field Fq2(t)Fq2(t) defined by the equation y2+xy−tdy=x3y2+xy−tdy=x3. Its rank is q   when q≡1mod3 and its rank is q−2q−2 when q≡2mod3. We describe an explicit method for producing points on this elliptic curve. In case q≢11mod12, our method produces points which generate a full-rank subgroup. Our strategy for producing rational points on EdEd makes use of a dominant map from the degree d   Fermat surface over Fq2Fq2 to the elliptic surface associated to EdEd. We in turn study lines on the Fermat surface FdFd using certain multiplicative character sums which are interesting in their own right. In particular, in the q≡7mod12 case, a character sum argument shows that we can generate a full-rank subgroup using μdμd-translates of a single rational point.